Vol. 1, No. 1, 2005

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On dimensional dual hyperovals ${\mathcal S}^{d+1}_{\sigma,\phi}$

Hiroaki Taniguchi and Satoshi Yoshiara

Vol. 1 (2005), No. 1, 197–219

A d-dimensional dual hyperoval Sσ,ϕd+1 inside PG(2d + 1,2) (d 2) was constructed by Yoshiara, for a generator σ of Gal(GF(q)GF(2)) and an o-polynomial ϕ(X) of GF(q)[X] (q = 2d+1). There, its automorphism group is determined and a criterion is given for these dimensional dual hyperovals to be isomorphic, assuming that the map ϕ on GF(q) induced by ϕ(X) lies in Gal(GF(q)GF(2)). In this paper, we extend these results for a monomial o-polynomial ϕ. We show that Aut(Sσ,ϕd+1)GL3(2) or Zq1.Zd+1 according as d = 2 or d 3, if ϕ(X) is monomial but ϕGal(GF(q)GF(2)). In particular, a special member X(0) of Sσ,ϕd+1 is always fixed by any automorphism of Sσ,ϕd+1. Furthermore, Sσ,ϕd+1Sσ,ϕd+1 if and only if either (σ,ϕ) = (σ,ϕ) or σσ = ϕϕ = id.

dimensional dual hyperoval, o-polynomial
Received: 13 January 2005
Accepted: 17 January 2005
Hiroaki Taniguchi
Satoshi Yoshiara